Given a countable residually finite group $\Gamma$, we write $\Gamma_n \to e$ if $(\Gamma_n)$ is a sequence of normal subgroups of finite index such that any infinite intersection of $\Gamma_n$'s contains only the unit element $e$ of $\Gamma$. Given a $\Gamma$-module $M$ we are interested in the multiplicative Euler characteristics \begin{equation} \label{eq:1a} \chi (\Gamma_n , M) = \prod_i |H_i (\Gamma_n , M)|^{(-1)^i} \end{equation} and the limit in the field $\mathbb Q_p$ of $p$-adic numbers \begin{equation} \label{eq:1b} h_p := \lim_{n\to\infty} (\Gamma : \Gamma_n)^{-1} \log_p \chi (\Gamma_n , M) \; . \end{equation} Here $\log_p : \mathbb Q^{\times}_p \to \mathbb Z_p$ is the branch of the $p$-adic logarithm with $\log_p (p) = 0$. Of course, neither expression will exist in general. We isolate conditions on $M$, in particular $p$-adic expansiveness which guarantee that the Euler characteristics $\chi (\Gamma_n,M)$ are well defined. That notion is a $p$-adic analogue of expansiveness of the dynamical system given by the $\Gamma$-action on the compact Pontrjagin dual $X=M^∗$ of $M$. Under further conditions on $\Gamma$ we also show that the renormalized $p$-adic limit in the second formula exists and equals the $p$-adic $R$-torsion of $M$. The latter is a $p$-adic analogue of the Li–Thom L2 $R$-torsion of a $\Gamma$-module $M$ which they related to the entropy h of the $\Gamma$-action on $X$. We view the limit $h_p$ as a version of entropy which values in the $p$-adic numbers and the equality with $p$-adic $R$-torsion as an analogue of the Li–Thom formula in the expansive case. We discuss the case $\Gamma = \mathbb{Z}^n$ in more detail where our theory is related to Serre's intersection numbers on arithmetic schemes.

中文翻译：

---

重整对数欧拉特征的$ p $ -Adic极限

给定一个可数的残差有限组$ \ Gamma $，如果$（\ Gamma_n）$是一个有限索引的普通子组序列，使得$ \ Gamma_n $的任何无限交集仅包含$ \ Gamma $，我们将$ \ Gamma_n \写入e $ $ \ Gamma $的单位元素$ e $。给定一个$ \ Gamma $模块$ M $，我们对乘法欧拉特性\ begin {equation} \ label {eq：1a} \ chi（\ Gamma_n，M）= \ prod_i | H_i（\ Gamma_n，M）感兴趣| ^ {（-1）^ i} \ end {equation}以及$ p $ -adic数\ begin {equation} \ label {eq：1b}的字段$ \ mathbb Q_p $的限制h_p：= \ lim_ {n \ to \ infty}（\ Gamma：\ Gamma_n）^ {-1} \ log_p \ chi（\ Gamma_n，M）\; 。\ end {equation}这里$ \ log_p：\ mathbb Q ^ {\ times} _p \ to \ mathbb Z_p $是$ p $ -adic对数的分支，其中$ \ log_p（p）= 0 $。当然，这两种表达一般都不会存在。我们隔离$ M $的条件，尤其是$ p $ -adic膨胀性，它确保欧拉特性$ \ chi（\ Gamma_n，M）$得到很好的定义。这个概念是动态系统扩展性的$ p $ -adic类似物，它由紧凑型Pontrjagin对偶$ X = M ^ * $上的$ \ Gamma $动作给出。在$ \ Gamma $的进一步条件下，我们还显示出第二个公式中经过重新规范化的$ p $ -adic极限存在，并且等于$ M $的$ p $ -adic $ R $扭转。后者是$ \ Gamma $模块$ M $的Li–Thom L2 $ R $扭转的$ p $ -adic类似物，它们与$ X的$ \ Gamma $作用的熵h有关。 $。我们将极限$ h_p $看作是熵的一个版本，它的价值为$ p $ -adic值，而与$ p $ -adic $ R $ -torsion相等的值在扩展情况下类似于Li–Thom公式。